The Federalist and Republican parties of old share a number of important similarities with the modern-day Democratic and Republican parties.
The Federalists were similar to today's Republicans in that they were very much the party of business and the East Coast banking elite. Having said that, it's fair to say that the Federalists would not have had much time for the kind of populist rhetoric that has become the modern-day GOP's stock in trade since Donald Trump became the party's candidate for the 2016 presidential election.
Such populist rhetoric was very much a feature of the old Democratic-Republican Party under Thomas Jefferson. Jefferson presented his party as being on the side of the little guy, small farmers and businessmen whose livelihoods were supposedly threatened by the greed of East Coast bankers.
Jefferson's party was similar to today's Republican Party in that it advocated limited government. To that end, it sought to protect states' rights as much as possible as a bulwark against what it saw as the overreach of the federal government.
The Federalists of old were similar to the modern-day Democratic Party in that they were supporters of a strong central government. The Federalists, no less than the Democrats of the day, tended to see the federal government as a powerful weapon for good.
Of course, the Federalists had different aims than present-day Democrats. They looked upon a strong central government primarily as a means to the establishment of the necessary regulatory regime for a stable banking system and a prosperous economy.
The Democrats of today, on the other hand, see the federal government as providing the sole means of attaining a wide variety of specific social and economic outcomes, such as an increased minimum wage and higher taxes on the rich.